A060890 -id:A060890 - cp's OEIS frontend

A269442 a(n) = n(n^8 + 1)(n^4 + 1)(n^2 + 1)(n + 1) + 1.

1, 17, 131071, 64570081, 5726623061, 190734863281, 3385331888947, 38771752331201, 321685687669321, 2084647712458321, 11111111111111111, 50544702849929377, 201691918794585181, 720867993281778161, 2345488209948553531, 7037580381120954241

Offset: 0

Ilya Gutkovskiy, Feb 26 2016

a(n) = Phi_17(n) where Phi_k(x) is the k-th cyclotomic polynomial.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. similar sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), this sequence (k=17), A060891 (k=18), A269446 (k=19).

List([0..20], n-> n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1); # G. C. Greubel, Apr 24 2019

[n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1: n in [0..20]]; // Vincenzo Librandi, Feb 27 2016

Mathematica
```
Table[Cyclotomic[17, n], {n, 0, 15}]
```

a(n)=n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1 \\ Charles R Greathouse IV, Jul 26 2016

[n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1 for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 +130918*x^2 +62343506*x^3 +4646748160*x^4 +102074708252*x^5 +878064150546*x^6 +3419813860214*x^7 +6502752956958*x^8 +6232856389160*x^9 +3004612851498*x^10 +701875014878*x^11 +73106078368*x^12 +2893069436*x^13 +31542430*x^14 +43674*x^15 +x^16)/(1 - x)^17.

Sum_{n>=0} 1/a(n) = 1.05883117453...

A102909 a(n) = Sum_{j=0..8} n^j.

Original entry on oeis.org

1, 9, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111, 235794769, 469070941, 883708281, 1589311291, 2745954241, 4581298449, 7411742281, 11668193551, 17927094321, 26947368421, 39714002329, 57489010371, 81870575521, 114861197401

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Mar 01 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A001016, A060890, A059839.

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), this sequence (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..8]]): n in [0..30]]; // G. C. Greubel, Feb 13 2018

1 + Sum[Range[0, 30]^j, {j, 1, 8}] (* G. C. Greubel, Feb 13 2018 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,9,511,9841,87381,488281,2015539,6725601,19173961},30] (* Harvey P. Dale, Feb 01 2025 *)

a(n)=n^8+n^7+n^6+n^5+n^4+n^3+n^2+n+1 \\ Charles R Greathouse IV, Oct 07 2015

[sum(n^j for j in (0..8)) for n in (0..30)] # G. C. Greubel, Apr 14 2019

a(n) = (n^2+n+1) * (n^6+n^3+1) and so is never prime. - Jonathan Vos Post, Dec 21 2012
G.f.: (x^8 + 162*x^7 + 3418*x^6 + 14212*x^5 + 16578*x^4 + 5482*x^3 + 466*x^2 + 1)/(1-x)^9. - Colin Barker, Nov 05 2012, edited by M. F. Hasler, Dec 31 2012
a(n) = (n^9-1)/(n-1) with a(1) = 9. - L. Edson Jeffery and M. F. Hasler, Dec 30 2012
E.g.f.: exp(x)*(1 + 8*x + 247*x^2 + 1389*x^3 + 2127*x^4 + 1206*x^5 + 288*x^6 + 29*x^7 + x^8). - Stefano Spezia, Oct 03 2024

Offset corrected by N. J. A. Sloane, Dec 30 2012

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A258805 Primes of the form k^8 + 1.

Original entry on oeis.org

2, 257, 65537, 37588592026706177, 92170395205042177, 147578905600000001, 284936905588473857, 3503536769037500417, 11007531417600000001, 11763130845074473217, 47330370277129322497, 50024641296100000001, 76872571987558646017, 416806419029812551937

Offset: 1

Vincenzo Librandi, Jun 11 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002496, A037896.

Cf. A006314 (associated n), A060890.

[a: n in [1..500] | IsPrime(a) where a is n^8+1];

Mathematica
```
Select[Range[500]^8 + 1, PrimeQ]
```

is(n)=ispower(n-1,8) && isprime(n) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A060890(A006314(n)). - Michel Marcus, Jun 11 2015

A258806 a(n) = n^7 + 1.

Original entry on oeis.org

1, 2, 129, 2188, 16385, 78126, 279937, 823544, 2097153, 4782970, 10000001, 19487172, 35831809, 62748518, 105413505, 170859376, 268435457, 410338674, 612220033, 893871740, 1280000001, 1801088542, 2494357889, 3404825448, 4586471425, 6103515626, 8031810177

Offset: 0

Vincenzo Librandi, Jun 11 2015

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Subsequence of A004864.
Sequences of the type n^k+1: A002522 (k=2), A001093 (k=3), A002523 (k=4), A002561 (k=5), A002604 (k=6), this sequence (k=7), A060890 (k=8).
Cf. A300785.

List([0..30],n->n^7+1); # Muniru A Asiru, Oct 24 2018

Magma
```
[n^7+1: n in [0..40]];
```

I:=[1,2,129,2188, 16385,78126,279937,823544]; [n le 8 select I[n] else 8*Self(n-1) - 28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) -28*Self(n-6) + 8*Self(n-7)-Self(n-8): n in [1..40]];

seq(n^7+1,n=0..30); # Muniru A Asiru, Oct 24 2018

Table[n^7 + 1, {n, 0, 40}] (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 2, 129, 2188, 16385, 78126, 279937, 823544}, 40]

a(n)=n^7+1 \\ Charles R Greathouse IV, Jun 11 2015

[n^7+1 for n in (1..40)] # Bruno Berselli, Jun 11 2015

G.f.: (1 - 6*x + 141*x^2 + 1156*x^3 + 2451*x^4 + 1170*x^5 + 127*x^6)/(1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
a(n) = (n + 1)*(n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
a(n) = Sum_{k=0..n} A300785(n,k). - Kolosov Petro, Oct 23 2018
E.g.f.: (1 +x +63*x^2 +301*x^3 +350*x^4 +140*x^5 +*21*x^6 +x^7)*exp(x). - G. C. Greubel, Oct 24 2018

A194003 Number of prime factors of n^8 + 1, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 3, 2, 6, 2, 4, 3, 3, 2, 2, 2, 4, 3, 3, 2, 4, 6, 3, 2, 2, 4, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 5, 2, 3, 2, 4, 4, 4, 3, 6, 2, 5, 2, 2, 2, 5, 2, 5, 4, 4, 3, 4, 3, 5, 4, 2, 3, 4, 2, 4

Offset: 0

Jonathan Vos Post, Aug 10 2011

This is to A193330 as A002523(n) = n^4+1 is to A002522(n) = n^2 + 1, and as A060890(n) = n^8+1 is to A002522(n) = n^2 + 1. a(n) = 1 when n^8+1 is prime, iff n is in {1, 2, 4} unless there is a larger Fermat prime than 65537.

			a(10) = 2 because 10^8 + 1 = 100000001 = 17 * 5882353 has 2 prime factors.
a(40) = 6 because 40^8 + 1 = 6553600000001 = 17^2 * 113 * 337 * 641 * 929 has 6 prime factors (with multiplicity) and is the smallest example not squarefree.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A060890, A002523, A193432, A193562, A193929.

[0] cat [&+[p[2]: p in Factorization(n^8+1)]:n in [1..90]]; // Marius A. Burtea, Feb 09 2020

Join[{0}, Table[Total[Transpose[FactorInteger[n^8 + 1]][[2]]], {n, 50}]]
PrimeOmega[Range[0,90]^8+1] (* Harvey P. Dale, May 27 2018 *)

a(n) = bigomega(n^8+1); \\ Michel Marcus, Feb 09 2020

a(n) = A001222(A060890(n)) = bigomega(n^8+1) or Omega(n^8+1)

A235983 Numbers n of the form p^8 + 1 (for prime p) such that n^8 + 1 is also prime.

Original entry on oeis.org

5764802, 146830437604322, 498311414318121122, 554786279519086052117762, 830149880552636599409282, 12628864335244989661982882, 33144490094099439467757602, 47203563969247823515902242, 179357590196404221918909122, 397370823547272766854136322, 2043714066708245412886790402, 139717795608648816763227344162

Offset: 1

Derek Orr, Jan 17 2014

nonn

All numbers are congruent to 2 mod 20.

			33144490094099439467757602 = 1549^8 + 1 (1549 is prime) and 33144490094099439467757602^8 + 1 is prime, so 33144490094099439467757602 is a member of this sequence.

Cf. A060890, A006314.

Select[Prime[Range[600]]^8+1,PrimeQ[#^8+1]&] (* Harvey P. Dale, Dec 21 2014 *)

import sympy
from sympy import isprime
{print(n**8+1) for n in range(10000) if isprime(n) if isprime((n**8+1)**8+1)}

A354053 Decimal expansion of Sum_{k>=0} 1 / (k^8 + 1).

Original entry on oeis.org

1, 5, 0, 4, 0, 6, 2, 1, 3, 3, 3, 1, 4, 7, 9, 9, 5, 1, 1, 2, 9, 2, 9, 0, 5, 4, 1, 7, 4, 5, 1, 1, 2, 7, 0, 7, 5, 2, 4, 5, 4, 1, 4, 3, 6, 3, 8, 2, 0, 3, 5, 1, 9, 7, 5, 4, 5, 8, 6, 3, 5, 3, 5, 7, 8, 1, 8, 8, 1, 2, 6, 9, 5, 1, 6, 4, 5, 6, 6, 3, 3, 4, 0, 7, 2, 0, 0, 6, 6, 1, 3, 9, 8, 5, 1, 6, 8, 4, 2, 8, 1, 8, 2, 4, 3

Offset: 1

Vaclav Kotesovec, May 16 2022

			1.504062133314799511292905417451127075245414363820351975458635357818812...

Cf. A060890, A113319, A354051, A354052.

evalf(1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8, 100);

RealDigits[Chop[N[Sum[1/(k^8 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^8 + 1))
```

Equals 1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8.

Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(8*k)-1). - Amiram Eldar, May 20 2022

A258809 a(n) = n^8 - 1.

Original entry on oeis.org

0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, 99999999, 214358880, 429981695, 815730720, 1475789055, 2562890624, 4294967295, 6975757440, 11019960575, 16983563040, 25599999999, 37822859360, 54875873535, 78310985280, 110075314175

Offset: 1

Vincenzo Librandi, Jun 11 2015

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A024006, A060890.

Cf. similar sequences listed in A258807.

Magma
```
[n^8-1: n in [1..40]];
```

Table[n^8 - 1, {n, 33}] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720}, 40]

G.f.: x^2*(225 + 4535*x + 14595*x^2 + 18069*x^3 + 569*x^4 + 3999*x^5 - 2511*x^6 + 1079*x^7 - 270*x^8 + 30*x^9) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
a(n) = (n - 1)*(n + 1)*(n^2 + 1)*(n^4 + 1) = -A024006(n). [Bruno Berselli, Jun 12 2015]

A326618 a(n) = n^18 + n^9 + 1.

Original entry on oeis.org

1, 3, 262657, 387440173, 68719738881, 3814699218751, 101559966746113, 1628413638264057, 18014398643699713, 150094635684419611, 1000000001000000001, 5559917315850179173, 26623333286045024257, 112455406962561892503, 426878854231297789441, 1477891880073843750001

Offset: 0

Richard N. Smith, Jul 15 2019

nonn

a(n) = Phi_27(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Richard N. Smith, Table of n, a(n) for n = 0..2000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument

Sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A269442 (k=17), A060891 (k=18), A269446 (k=19), A060892 (k=20), A269483 (k=21), A269486 (k=22), A060893 (k=24), A269527 (k=25), A266229 (k=26), this sequence (k=27), A270204 (k=28), A060894 (k=30), A060895 (k=32), A060896 (k=36).

Cf. A153440 (indices of prime terms).

[n^18+n^9+1: n in [0..17]]; // Vincenzo Librandi, Jul 15 2019

Table[n^18 + n^9 + 1, {n, 0, 17}] (* Vincenzo Librandi, Jul 15 2019 *)
Table[Cyclotomic[27, n], {n, 0, 17}]

a(n) = polcyclo(27, n); \\ Michel Marcus, Jul 20 2019

cp's OEIS Frontend

A269442 a(n) = n*(n^8 + 1)*(n^4 + 1)*(n^2 + 1)*(n + 1) + 1.

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A102909 a(n) = Sum_{j=0..8} n^j.

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A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

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A258805 Primes of the form k^8 + 1.

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A258806 a(n) = n^7 + 1.

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A194003 Number of prime factors of n^8 + 1, counted with multiplicity.

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A235983 Numbers n of the form p^8 + 1 (for prime p) such that n^8 + 1 is also prime.

A269442 a(n) = n(n^8 + 1)(n^4 + 1)(n^2 + 1)(n + 1) + 1.